L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.499 − 0.866i)6-s + (1.73 + i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (−3 − 5.19i)11-s − 0.999i·12-s + (−0.866 − 3.5i)13-s + 1.99·14-s + (−0.5 − 0.866i)16-s + (−2.59 − 1.5i)17-s − 0.999i·18-s + (1 − 1.73i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.204 − 0.353i)6-s + (0.654 + 0.377i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.904 − 1.56i)11-s − 0.288i·12-s + (−0.240 − 0.970i)13-s + 0.534·14-s + (−0.125 − 0.216i)16-s + (−0.630 − 0.363i)17-s − 0.235i·18-s + (0.229 − 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.622609081\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.622609081\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (0.866 + 3.5i)T \) |
good | 7 | \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-6.06 + 3.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.66 - 5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 13iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.1 - 7i)T + (48.5 + 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.858285094046218973531638538230, −8.027302032877106725705650680745, −7.58913894123158457657966380171, −6.32502173584417055060241410384, −5.57888720664673909398696779439, −4.95288303532525647942837085101, −3.73319036892044647069089576359, −2.90456642338957148776161984674, −2.15674617584918222112873756750, −0.67890398560157800345575475603,
1.90065909901455752680545245191, 2.53813844881601110681485515356, 4.14797206687452650102444477767, 4.32541736153463609214917158232, 5.23969898136985197182475183324, 6.30630340592177263819999125436, 7.26641606489907442089947334043, 7.72113394579276720093790959761, 8.500976459042391994726312606068, 9.481172995103403417215197102958